This problem reformulates (in matrix form) the Jacobi iterative method for solving linear systems of equations and provides a justification for its convergence criterion. (a) Split the matrix A in...

This problem reformulates (in matrix form) the Jacobi iterative method for solving linear systems of equations and provides a justification for its convergence criterion. (a) Split the matrix A in terms of a lower triangular matrix, a diagonal matrix, and an upper triangular matrix: A = D(L + I + U) (4.323) Now, if you are told that the following relation: x(k) = Bx(k-1) + c (4.324) is fully equivalent to the Jacobi method of Eq. (4.169), express B and c in terms of known quantities. (Unsurprisingly, a similar identification can be carried out for the Gauss–Seidel method as well.) (b) We will now write the matrix relation Eq. (4.324) for the exact solution vector x. This is nothing other than: x = Bx + c (4.325) Combine this with Eq. (4.324) to relate x(k) - x with x(k) - x(k-1). Discuss how small B has to be for x(k)-x(k-1) to constitute a reasonable correctness criterion. (c) Examine the infinity norm B8 and see what conclusion you can reach about its magnitude for the case where A is diagonally dominant.